 # Combinations, Sets for Frequency System: Lotto 5/39, Lotto 6/49, Pick-3, Pick-4

## By Ion Saliu, Founder of: Lottery Mathematics, Lotto Programming Science  ### Jackpot Lotto: Calculate Combinations from 3 Groups of NumbersPick 3 4 Lotteries: Calculate Sets from 3 Groups of NumbersPick 5 Lottery: Calculate ALL Sets in Quinto LotteryResources in Lottery Software, Strategies, Lotto Systems

Special lottery software: SkipDecaFreq, programs in the Bright / Ultimate application bundles. ## Jackpot Lotto: Calculate Combinations from 3 Groups of Numbers

First capture by the WayBack Machine (web.archive.org) February 23, 2007.

We divided the lotto numbers (jackpot games) in 3 groups of N1, N2, and N3 numbers. We work here with 2 lotto games that draw K numbers: K = 5 and K = 6. The lotto numbers are unique in each group: There is no overlap. We apply the combination formula to each group and finally multiply the 3 results.

C(N, M) is the general formula for combinations of N numbers taken M at a time.

N*(N-1)*(N-2)*(N-3)* ... *(N-M+1)
C(N, M) = -----------------------------------------------
1*2*3* ... *M

• Example: 5/39 lotto, with the following contents: Group #1 consists of 5 lotto numbers; group #2 has a total of 14 numbers; group #3 consists of 20 lotto numbers (total: 39 numbers). The frequency distribution is 1  1  3 (must total 5 as it is a 5-number lotto game; must sum up to 6 for 6-number lotto games):
• Group #1 combinations: C(N1, K1) = C(5, 1) = 5
• Group #2 combinations: C(N2, K2) = C(14, 1) = 14
• Group #3 combinations: C(N3, K3) = C(20, 3) = 1140
• Total combosnations (a favorite of mine!): 5 * 14 * 1140 = 79800.

The following tables show the amount of lottery combinations from 3 groups of numbers ordered by frequency. Lotto games analyzed: 5/39 and 6/49.

### Lotto 5/39 Combinations for Frequency System

 Group_1 = 5 #s Group_2 = 14 #s Group_3 = 20 #s Total Combos 0 0 5 15504 0 5 0 2002 5 0 0 1 0 1 4 67830 0 4 1 20020 1 0 4 24225 1 4 0 5005 4 0 1 100 4 1 0 70 0 2 3 103740 0 3 2 69160 2 0 3 11400 2 3 0 3640 3 0 2 1900 3 2 0 910 1 1 3 79800 1 3 1 36400 3 1 1 2800 1 2 2 86450 2 1 2 26600 2 2 1 18200 Total C(39, 5) ~ ~ 575757 ### Lotto 6/49 Combinations for Frequency System

 Group_1 = 6 #s Group_2 = 18 #s Group_3 = 25 #s Total Combos 0 0 6 177100 0 6 0 18564 6 0 0 1 0 1 5 956340 0 5 1 214200 1 0 5 318780 1 5 0 51408 5 0 1 150 5 1 0 108 0 2 4 1935450 0 4 2 918000 2 0 4 189750 2 4 0 45900 4 0 2 4500 4 2 0 2295 1 1 4 1366200 1 4 1 459000 4 1 1 6750 0 3 3 1876800 3 0 3 46000 3 3 0 16320 1 2 3 2111400 1 3 2 1468800 2 1 3 621000 2 3 1 306000 3 1 2 108000 3 2 1 76500 2 2 2 688500 Total C(49, 6) ~ ~ 13983816 ## Pick Lotteries: Calculate Sets from 3 Groups of Digits

Although the odds are much lower, calculations in pick (digit, daily games) lotteries are more complex. The sets in these games have not only unique digits (as in jackpot lotto), but also repeat digits (doubles, triples, quadruples).

### I. Calculating total unique sets is the easiest part. Total digits: N = 10; digits per set M = 3 (pick-3) or M = 4 (pick-4):

• Total unique M of N straight sets = (M! * C(N, M))
• Total single sets pick-3 = 3! * 120 = 720
• Total single sets pick-4 = 4! * 210 = 5040.

### II. Calculating total duplicate sets is more complicated and involves more steps. The calculations are different from one type of duplication to another (doubles, triples, quadruples, quads).

• Pick-3 doubles  All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with each of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple); thus, the second element of the calculation is 10 * 9 = 90. A 3-digit set that contains a double and a single can be arranged in C(3, 2) = 3 ways; the final term of the calculation is 90 * 3 = 270 total doubles.

• Pick-4 triples  All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with each of the 9 remaining digits (we do not combine a triple digit with itself as it would result in a quadruple); thus, the second element of the calculation is 10 * 9 = 90. A 4-digit set that contains a triple and a single can be arranged in C(4, 3) = 4 ways; the final term of the calculation is 90 * 4 = 360 total triples.

• Pick-4 1 double + 2 singles  All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with 2 of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple or a quadruple): 9 * 8 = 72; thus, the second element of the calculation is 10 * 72 = 720. A 4-digit set that contains a double and 2 singles can be arranged in C(4, 2) = 6 ways; the final term of the calculation is 720 * 6 = 4320 total 1 double + 2 singles sets.

• Pick-4 2 doubles  All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with doubles of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple or a quadruple); thus, the second element of the calculation is 10 * 9 = 90. A 4-digit set that contains 2 distinct doubles can be arranged in C(4-1, 1) = 3 ways (e.g. 0011, 0101, 0110). The final term of the calculation is 90 * 3 = 270 total 2 doubles sets.

• Pick 3 triples  Each of the 10 digits forms a triple: 10 sets.
• Pick 4 quadruples  Each of the 10 digits forms a quad: 10 sets.

• Pick 3 Boxed: 120 (singles) + 90 (doubles) + 10 (triples) = 220 sets.
• Pick 4 Boxed: 210 (singles) + 720/2 (1 double + 2 singles) + 90/2 (2 doubles) + 90 (triples) + 10 (quads) = 715 sets.

### Pick-3 Sets for Frequency System

 Group_1 2 digits Group_2 3 digits Group_3 5 digits Single Sets Double Sets Triple Sets Total Sets 0 0 3 60 60 5 125 0 3 0 6 18 3 27 3 0 0 0 6 2 8 0 1 2 180 45 0 225 0 2 1 90 45 0 135 1 0 2 120 30 0 150 1 2 0 36 18 0 54 2 0 1 30 30 0 60 2 1 0 18 18 0 36 1 1 1 180 0 0 180 Total Pick-3 ~ ~ 720 270 10 1000 ### Pick-4 Sets for Frequency System

 Group_1 2 digits Group_2 3 digits Group_3 5 digits Single Sets 1-Pair Sets 2-Pair Sets Triple+1 Sets Quad Sets Total Sets 0 0 4 120 360 60 80 5 625 0 4 0 0 36 18 24 3 81 4 0 0 0 0 6 8 2 16 0 1 3 720 720 0 60 0 1500 0 3 1 120 360 0 60 0 540 1 0 3 480 480 0 40 0 1000 1 3 0 48 144 0 24 0 216 3 0 1 0 120 0 40 0 160 3 1 0 0 72 0 24 0 96 1 1 2 1440 360 0 0 0 1800 1 2 1 720 360 0 0 0 1080 2 1 1 360 360 0 0 0 720 0 2 2 720 540 90 0 0 1350 2 0 2 240 300 60 0 0 600 2 2 0 72 108 36 0 0 216 Total Pick-4 ~ ~ 5040 4320 270 360 10 10000 ## Pick 5 Lottery: Calculate ALL Sets in Quinto Lottery

Axiomatic one, I do not have a lot of software for pick 5 digit (Quinto) lottery. Therefore I will not divide the pick-5 digits in groups based on frequency. I will calculate only the total sets: Singles, 1 double + 3 singles, 2 doubles + 1 single, 1 triple + 2 singles, 1 triple + 1 double, 1 quadruple + 1 single, quintets.

• Total unique M of N straight sets = (M! * C(N, M))
• Total single sets pick-5 = 5! * 252 = 30240.

• 1 double + 3 singles  All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with 3 of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple, a quadruple, or a quintet): 9 * 8 * 7 = 504; thus, the second element of the calculation is 10 * 504 = 5040. A 5-digit set that contains a double and 3 singles can be arranged in C(5, 2) = 10 ways; the final term of the calculation is 5040 * 10 = 50400 total 1 double + 3 singles sets.

• 2 doubles + 1 single (the most complicated case scenario)  All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with each of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple, a quadruple, or a quintet); thus, the second element of the calculation is 10 * 9 = 90. The 2 doubles are combined with each of the 8 remaining digits. A 5-digit set that contains 2 distinct doubles can be arranged in C(5, 2) = 10 ways; we also add to it the C(5, 1) = 5 of the single combinations: 10 + 5 = 15. The final term of the calculation is 90 * 8 * 15 = 10800 total 2 doubles + 1 single sets.

• 1 triple + 2 singles  All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with 2 of the 9 remaining digits; thus, the second element of the calculation is 9 * 8 = 72; 72 * 10 = 720. A 5-digit set that contains a triple and 2 singles can be arranged in C(5, 3) = 10 ways; the final term of the calculation is 720 * 10 = 7200 total 1 triple + 2 singles.

• 1 triple + 1 double  All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with each of the 9 doubles; thus, the second element of the calculation is 10 * 9 = 90. A 5-digit set that contains 1 triple and 1 double can be arranged in C(5, 3) = 10 ways; the final term of the calculation is 90 * 10 = 900 total 1 triple + 1 double.

• 1 quadruplet + 1 single  All 10 digits can form a quad; therefore the first term is 10. Each of the 10 quadruples is combined with each of the 9 remaining digits; thus, the second element of the calculation is 10 * 9 = 90. A 5-digit set that contains a quad and 1 single can be arranged in C(5, 4) = 5 ways; the final term of the calculation is 90 * 5 = 450 total 1 quadruplet + 1 single.

• Pick 5 quintets  Each of the 10 digits forms a quintuple: 10 sets.

• Grand total: 30240 + 50400 + 10800 + 7200 + 900 + 450 + 10 = 100,000 pick5 straight sets.

• Pick 5 Boxed: 252 (singles) + 840 (1 double + 3 singles) + 360 (2 doubles + 1 single) + 360 (1 triple + 2 singles) + 90 (1 triple + 1 double) + 90 (1 quad + 1 single) + 10 (quintets) = 2002 sets. ### All Pick-5 Lottery Sets: Distinct and Duplicate

 Singles 1 Double + 3 Singles 2 Doubles + 1 Single 1 Triple + 2 Singles 1 Triple + 1 Double 1 Quad + 1 Single Quintets TOTAL 30240 50400 10800 7200 900 450 10 100000 ### Boxed Pick-5 Lottery Sets

 Singles 1 Double + 3 Singles 2 Doubles + 1 Single 1 Triple + 2 Singles 1 Triple + 1 Double 1 Quad + 1 Single Quintets TOTAL 252 840 360 360 90 90 10 2002 ## Resources in Lottery Software, Strategies, Lotto Systems | Home | Search | New Writings | Odds, Generator | Contents | Forums | Sitemap |